Curve Forecast with the SOM Algorithm : Using a Tool to Follow the Time on a Kohonen Map P. ROUSSET

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1 Centre de Recherche S A M O S M A I S S E CNRS, UMR 8595 Statistiqe Appliqée & Modélisation Stochastiqe Crve Forecast with the SOM Algorithm : Using a ool to Follow the ime on a Kohonen Map P. ROUSSE Prépblication d SAMOS N 2 Janvier 2000

2 Crve Forecast with the SOM Algorithm : Using a ool to Follow the ime on a Kohonen Map Rosset Patrick Matisse-Samos UMR CNRS 8595 Université de Paris 90 re de olbiac, PARIS CEDEX 3 rosset@niv-paris.fr Abstract. o forecast a complete crve, we propose a method that consists in predicting from a rle based on a classification, which takes the present time class into accont. his techniqe is simpler than a vectorial prediction and solves some problems of long term forecasting. A type of error, which belongs to this method, imposes to take care of it. he SOM and a tool of visalization that permits to follow the time on that kind of classification give s a way to control it. he application to the polish electrical consmption is presented. Introdction A previos paper [5,3] explained that the prediction of a crve with the same qality for all vales is a problem for recrsive and vectorial methods. I recrsive methods, as each predicted vale is presented to compte its ftre, the estimation becomes worth and worth as the ftre is far. he behavior of series created with a mltilayer perceptron recrrence eqation is chaotic and strongly dependant on the initial vales. So a mltilayer perceptron cannot be sed in long term forecasting ([0,2,3,4]). he vectorial methods forecast all the crve vales at the same time and with the same qality bt wold have to take into accont the dependence of vales, and lead to very costly algorithms. A method that ses a classification was firstly proposed in [6]. he one that we propose in [5,4,5] gives the same importance to the complete information to forecast the ftre. We propose here the same type of approach bt considering the present information. We se the example of Polish Electrical Consmption as a spport to present the method. We thank Professor Osowski from Warsaw echnical University for giving s at disposal these data. 2 he Method: Let s consider a time series X(t, i), with a doble indexation. he time is denoted by (t, i), where t is the slow scale (the day for example), and i is the rapid one (the hor for example). For each t {,, the second index takes its vales between and I, where I does not depend on t. he sccessive vales are groped according to the t index, and we set X(t) = (X(t, i), i =,..., I)

3 We first make a classification of all the crves X (t) into U classes C, each one being represented by its centroid G. For each class C, we can define the empirical probability P (') for the ftre crve X(t+) of an individal X(t) of C to be classified in the class C '. { X t C, X t+ C P ( ' ) =, where {proposition is or 0 when the ' { X C t proposition is tre or false. hen the estimation of the complete crve X(t+) when X(t) is classified in the class is given by: U X ˆ ( t + ) = P ( ' ) () ' = G ' Xˆ ( t + ) can be seen as a barycenter of some convenient G. 3 A Problem of this echniqe: his approach sffers a typical problem illstrated by the following example. Sppose that we have 3 classes C, C 2, C 3 represented by the vectors G =(, 0), G 2 =(/2, /2) G 3 =(0, ). he following table shows the repartition of the 3 class ftres. C ftre is C or C 3, bt never C 2. X(t)\X(t+) C C 2 C 3 C 50% 0% 50% C 2 0% 00% 0% C 3 50% 0% 50% he rle () predicts the ftre of the element in C by ˆ X ( t + ) = G + G3 =, = G and X ˆ ( t + ) is classified in C 2, which is contradictory with the previos table. 4 Soltion of the Problem: he Neighborhood Notion in the SOM. If vectors G in expression () are neighbor, the barycenter X ˆ ( t + ) is closed to each of them. At the contrary, if they are far from each other, X ˆ ( t + ) is far from them. he Kohonen algorithm [7,8,9,,2] defines a neighborhood strctre between the classes that we can se to solve the previos problem.

4 A tool to follow the time on the SOM map: presentation of the polish electrical consmption We present the application of the forecasting rle () and the tool to follow the ftre on a Kohonen map in a real example: he horly Polish Power (expressed in Mwh) from 0/0/986 to 3/2/994. his data set was kindly lent by Pr. Osowski from Warsaw echnical University. In a precedent stdy [4,5], we jstified to separate the daily information Y(t) into three parts, the mean m(t), the standard deviation σ(t) and the profile X(t). So we had the relation: Y(t)= m(t) + σ(t) X(t) he profile was predicted with a rle issed from a classification and the two other parameters were forecasted with classical methods sch as ARMA models or Mltilayer Perceptron. Here we are going to se the rle () to predict the ftre of the profiles. We consider a cylindrical Kohonen 0 x 0 network, where left and right borders are neighbor. We train it with all the profiles. he first map shows the class centroids, the second all the crves of a class in its corresponding nit. So we can control if the vectors sed in expression () are neighbor. Now we need a tool to visalize where the (t+) day is classified when the (t) one belongs to class. Figre : NB= 38 NB= 47 NB= 49 NB= 2 NB= 26 NB= 5 NB= 40 NB= 59 NB= 6 NB= 30 NB= 39 NB= 2 NB= 25 NB= 3 NB= 28 NB= 0 NB= 88 NB= 32 NB= 45 NB= 2 NB= 26 NB= 5 NB= 44 NB= 25 NB= 20 NB= 5 NB= 29 NB= 45 NB= 26 NB= 42 NB= 4 NB= 50 NB= NB= 9 NB= 44 NB= 32 NB= 3 NB= 0 NB= 0 NB= 34 NB= 6 NB= 48 NB= 38 NB= 3 NB= 4 NB= 9 NB= 6 NB= 46 NB= 29 NB= 7 NB= 20 NB= NB= 4 NB= 24 NB= 4 NB= 0 NB= 8 NB= 28 NB= 28 NB= 40 NB= 48 NB= 75 NB= 9 NB= 4 NB= 7 NB= 20 NB= 2 NB= 3 NB= 33 NB= 33 NB= 22 NB= 3 NB= 8 NB= 24 NB= 33 NB= 7 NB= 9 NB= 35 NB= 42 NB= 43 NB= 47 NB= 35 NB= 33 NB= 25 NB= 22 NB= 22 NB= 4 NB= 4 NB= 42 NB= 37 NB= 66 NB= 24 NB= 33 NB= 42 NB= 47 NB= 36 NB= 0 NB= 2 NB= 48 NB= 52 Figre 2

5 he tool: Let s extract the individals of the class and bild the set E of their following days (let s take for example the nit 32). E = { X ( t + ) / X ( t) C he map in figre 3, called ftre of the class =32, represents the repartition of E into the U Kohonen classes. he nit ' is white, gray light, gray or black when an individal of E is classified in ' with the freqency respectively 0%, between 0% and 20%, between 20% and 40%, more than 40%. We can see that most of the nits are empty and the others are closed on the map. We can represent sch a figre for all the nits and we chose to arrange them on the Kohonen map, as in figre 4, in order to visalize all the class ftres and to be able to compare ones to others. 5. he Prediction of Polish Power. o explain the ftre, the ftre map is not sfficient (some areas are not connected). he figre 5 presented in [4,5,6], which projects the kind of day on the map, shows that hrsday and Friday have the same classification, bt not the same ftre, we have the same phenomenon with some Satrdays which are worked and classified as Monday to Friday. So, we have to introdce in the rle () the variable kind of day with 3 modalities k: Friday, Satrday and other days, as follow: P, k ' { X t C, t has the kind of day k, X t+ C = = t ( ' ) (') { X t C, t has the kind of day k t = Let s compare this approach (class of time t, kind of day) with the one presented in [4,5] (kind of month, kind of day) which rle does not give importance to what happens today to forecast tomorrow: Pm, k ( ' ) = t + ' { month of t is m, t has the kind of day k, X C { month of t is m, t has the kind of day k I By compting the error E, i i E = ( X Xˆ ) N 24 we obtain for the present method, when we got with the previos one. i= t 2 t 6. Conclsion: he error is abot the same bt we have to consider that the approaches are different. More than that, they can be mixed to have a prediction more precise or more

6 robst. he visalization of ftre solves the problem explained in section 3. As an evoltion, we can also introdce the information of the previos week as the (t-7) class. Figre 3 Figre 4 dimanche lndi mar->ven samedi Figre 5

7 Figre 6 : Forecasted and real crves for 04/02/87 Figre 7 : Forecasted and real crves for 04/02/97 to /02/87 References. Cottrell M., Fort J.C.: Etde d'n algorithme d'ato-organisation, Ann. Inst. Henri Poincaré, 23,, pp. -20 (987). 2. Cottrell M., Fort J.C., and Pages G.: heoretical aspects of the SOM algorithm, Nerocompting, 2, p9-38 (998) 3. Cottrell M., Girard B., Girard Y., Mller C. and Rosset P.: Daily Electrical Power Crves : Classification and forecasting Using a Kohonen Map From Natral to Artificial Neral Comptation, Proc. IWANN'95, J.Mira, F.Sandoval eds., Lectre Notes in Compter Science, Vol.930, Springer, 07-3, (995). 4. Cottrell M., Girard B., Girard Y., Mangeas M. and Mller C.: Neral modeling for time series: a statistical stepwise method for weight elimination, IEEE ransactions on Neral Networks, 6, 6, pp (995). 5. Cottrell M., Girard B. and Rosset P.: Forecasting of crves sing a Kohonen classification, Jornal of Forecasting, 7, (998)

8 6. Garcia ejedor A., Coscllela M., Bermejo C. and Montes R., A neral system for short-term load forecasting based on day-type classification, Proceedings ISAP 94, (994 ). 7. Kohonen.: Self-organized formation of topologically correct featre maps, Biol. Cybern., 43, pp (982). 8. Kohonen., Self-organization and Associative Memory, 3rd ed., Springer, New York (989). 9. Kohonen., Self-Organizing Maps, Springer, New York, 30 (995). 0. Lapedes, A. S. and Farber, R., Nonlinear signal processing sing neral networks: prediction and system modeling echnical Report, Los Alamos National Laboratory (987).. Mller, M., Cottrell, M., Girard, B., Girard, Y. and Mangeas, M.: A neral network tool for forecasting french electricity consmption, Proceedings of WCNN 94, San Diego,, pp (994). 2. Weigend A. S., Hberman B. A. and Rmelhart D. E., Predicting the ftre: a connectionist approach, Int. J. Neral Systems,, 3, pp (990). 3. Weigend, A. S. and Gershenfeld N. A., imes Series Prediction, Proceedings Volme XV, Santa Fe Institte, Reading, MA: Addison-Wesley (994). 4. Rosset P.: Application des algorithmes d'ato-organisation à la classification et à la prévision, hèse de l'niversité de Paris, sotene le 3 décembre Cottrell, M., Girard B., Rosset P.: Long term forecasting by combining Kohonen algorithm and standard prevision, Proc. of ICANN'97, Lasanne, Switzerland, PP (997) 6. Cottrell, M., Rosset P : he Kohonen algorithm: a powerfl tool for analysing and representing mltidimensional qantitative and qalitative data, Proc. of IWANN'97, biological and artificial comptation: from neroscience to technology, Lanzarote, Canary Island, Spain, pp (997)